# How is the frequency in the label behind microwave ovens determined? How could one measure the frequency?

It is very common online for people on youtube, forums, and other social media to bring up the following fun home experiment to "measure the speed of light" using chocolate and a microwave oven.

To explain it, the proposal is that by putting chocolate in a microwave (without the rotator plate), standing waves will form and leave heat marks and specific melting spots. You can measure the approximate distance between them to find the wavelength of the microwave in the oven (this works only for certain kinds of microwave ovens however). Then you can find the info at the back of the oven that details the frequency of the microwave. From this and your measurement, you can then compute the speed of light.

Now my worry here is, how exactly does the industry determine the frequency if the microwave? I have a sneaking suspicion that the people just calculate it from wavelength measurements and the accepted speed of light value. This renders the entire home experiment circular.

So I have two questions (an answer to either would be appreciated):

1. How do the people at the industry that design the microwave oven determine the frequency of the microwaves?
2. Is there a way to measure the frequency of the radiation emitted by the magnetron without presupposing anything about c? I am ok with a direct or indirect measurement as long as it avoids any mention of c.
• I imagine it can be calculated from the geometry of the magnetron. Commented Dec 31, 2022 at 2:46
• The microwave manufacturer doesn't care about the frequency as long as it stays within the 100 MHz-wide 2.4 GHz ISM band. I would be surprised if there wasn't 10s of MHz between units and as they warm up. So your home experiment is going to have a radio regulation-derived error of $\pm 0.05/2.45=\pm2\%$ Commented Dec 31, 2022 at 3:14
• @Hearth - doesn't that calculation presuppose a value for c though? Commented Dec 31, 2022 at 16:14
• @brhans You can get the speed of light if you trust Maxwell's equations and define a meter and second. That's going into metrology, you have to arbitrarily define something in your system of measurements. Previously, we defined a meter and a second and got the speed of light, since 1983, we define the speed of light and a second, and get a meter out. Commented Jan 1, 2023 at 20:07

It's worth noting that almost all home microwave ovens must operate within the ~2.4 GHz ISM band as that's the allocation of spectrum where they're allowed to operate for regulatory reasons. If they attempt to operate outside the ISM band allocated to them, they will interfere with important, possibly life-safety-critical signals.

This means that as long as you have enough trust that your microwave oven follows international standards, you already know that its frequency is around 2.45 GHz +/- 50 MHz without any further measurements.

1. The exact tools and techniques may very well vary from company to company. However, the magnetron is designed to a particular frequency (by tweaking geometry scaling as Hearth points out in a comment); doing a direct calculation from Maxwell's Equations would require knowing c.

As part of the design and testing phase, the output will almost certainly be measured with a suitable probe/attenuator and an RF spectrum analyzer (i.e. a specific piece of test equipment for this purpose).

2. You will need to take some definition of a second, but that definition does not need to come from c and distance. All you need are some caesium-133 atoms and the ability to observe the hyperfine transition. A second is defined as:

taking the fixed numerical value of the caesium frequency ΔνCs, the unperturbed ground-state hyperfine transition frequency of the caesium-133 atom, to be 9 192 631 770 when expressed in the unit Hz, which is equal to s–1.

You could directly trap some caesium-133 in a suitable trap and compare the radiation from its hyperfine transition to the radiation from your microwave, but this would be impractical and expensive for most applications. Instead, that reference is used to calibrate a chain of derived references, ending up with some kind of frequency reference available to you for your experiment.

Another way of getting a precise frequency reference would be to use a GPSDO, or GPS-Disciplined Oscillator. GPS satellites themselves carry caesium clocks and transmit precise timekeeping signals; by tracking those signals it's possible to obtain your own precise frequency reference.

Once you have a precise frequency reference (which we obtained without c or a length standard), you can measure the frequency of the microwave radiation. A conceptually simple way to measure the unknown microwave frequency would be to use fast counters: count the ticks of a 100 MHz reference clock until you reach 100 million, while counting the cycles of the microwave radiation. If everything is as expected, you'll count around 2.45 billion cycles of microwave radiation in that same time period.

On the other hand, a commercial RF analyzer does fairly complex downmixing using its local oscillator as well as further processing to provide a full spectrum (including both the main energy and any sidebands), which is too complex to explain in full here.

Note that in the end, you still need a definition of length to obtain your final value of c, and you run into a slight circular reasoning issue here, because the BIPM definition of a meter does use c. Without c, you don't have a ruler marked in cm, in, or any other convenient unit.

If you truly want to compute the speed of light without the need for c, you may be reduced to saying "x chocolate pieces per second" and using the chocolate piece itself as your ruler.

• The chart from your link shows there is no absorption peak at 2.4 GHz. The closest peak is for frozen water, slightly under 10 GHz, and at room temperature you'll need 20-30 GHz. There is certainly absorption. 2.4 GHz is a leftover from WWII. Commented Dec 31, 2022 at 4:18
• Microwave ovens work on the principal of Dielectric Heating, which has nothing specifically to do with water. Commented Dec 31, 2022 at 16:20
• Removed section, won't be near a computer to make a more substantial edit due to the holiday Commented Dec 31, 2022 at 16:22
• Microwave ovens deliberately do not run close to a water absorption peak since you want the skin depth of the radiation to be approximately the thickness of the object being heated, otherwise you'll burn the outside while the inside is still cold. I've heard that 900 MHz microwaves are actually much better at cooking food due to lower absorption, but are impractically large for most kitchens. Commented Dec 31, 2022 at 19:06

The people who design microwave ovens refer to the specifications of the magnetron.

For example, this Toshiba magnetron has an output frequency of 2450 +/-30MHz.

Some magnetrons are (mechanically) tunable but that's an unnecessary complexity in a microwave oven.

You could measure the frequency of the magnetron with a frequency counter. You can get such an instrument for as low as \$20, up to much more, depending on quality, performance etc. The basis of such a device is that the incoming frequency is digitally prescaled (divided down) to a frequency that is easy to count (perhaps tens of MHz) and then the counter is reset and allowed to count for a gate time of perhaps 100ms. The counting period is determined by mechanical oscillations of a quartz crystal and some digital circuitry. For example if you divide the frequency down to approximately 25MHz you could count for 0.1 second and get a count with 10Hz resolution if desired (6 digits). That's probably better than your crystal accuracy.